Foundations of entropy in complex systems

Jan Korbel

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arXiv · 2606.16312

Foundations of entropy in complex systems

Jan Korbel

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Abstract

This chapter reviews the foundations of entropy and their extensions to complex systems. We first discuss the relation between Boltzmann's formula, multiplicity, coarse-graining, and Shannon entropy, before introducing generalized entropies such as R\'enyi, Tsallis, and Burg entropy. We then examine Maxwell--Boltzmann, Bose--Einstein, and Fermi--Dirac statistics, structure-forming systems, sample-space reducing processes, P\'olya urns, and nonlinear dynamics. Axiomatic approaches are presented through the Shannon--Khinchin axioms, Tempesta group-composability, Hanel--Thurner asymptotic scaling, Shore--Johnson consistency axioms, and Lieb--Yngvason axioms. Finally, we discuss calibration invariance, Hanel--Thurner--Gell-Mann duality between linear and escort averages, and Kolmogorov--Nagumo averages, showing how the same distribution can arise from different entropies, constraints, or dynamics. These results emphasize that the choice of entropy should be guided by the structure and physical properties of the system.

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Jan Korbel. 2026-06-15. Foundations of entropy in complex systems. https://arxiv.org/abs/2606.16312

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